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Dense Minors in Graphs of Large Girth
 Combinatorica
"... this paper is to reduce the upper bound for the required girth to the correct order of magnitude: Theorem 1. For any inte k,e ve graph G of girth g(G) > 6 log k +3and #(G) # 3 has a minor H with #(H) >k. The best lower bound we have found is 8 3 log c, but we note that existing conjectu ..."
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conjectures about cubic graphs of large girth would raise this to about 4 log . Since an average degree of at least cr # log r forces a K r minor [ 5, 8 ], Theorem 1 has the following consequence: Corollary 2.The ee a constant c # R such thate ea graph G of girth g(G) # 6 log r<F1
Graph minors. X. Obstructions to treedecomposition
 J. COMB. THEORY, SERIES B
, 1991
"... Roughly, a graph has small “treewidth” if it can be constructed by piecing small graphs together in a tree structure. Here we study the obstructions to the existence of such a tree structure. We find, for instance: (i) a minimax formula relating treewidth with the largest such obstructions (ii) an ..."
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) an association between such obstructions and large grid minors of the graph (iii) a “treedecomposition” of the graph into pieces corresponding with the obstructions. These results will be of use in later papers.
Minors in Graphs of Large Girth
 J. Combin. Theory B
, 1988
"... We show that for every odd integer g 5 there exists a constant c such that every graph of minimum degree r and girth at least g contains a minor of minimum degree at least cr . This is best possible up to the value of the constant c for g = 5; 7 and 11. More generally, a wellknown conjecture ..."
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We show that for every odd integer g 5 there exists a constant c such that every graph of minimum degree r and girth at least g contains a minor of minimum degree at least cr . This is best possible up to the value of the constant c for g = 5; 7 and 11. More generally, a wellknown
Topological Minors in Graphs of Large Girth
 J. Combin. Theory B
, 1988
"... We prove that every graph of minimum degree at least r and girth at least 186 contains a subdivision of K_r+1 and that for r ≥ 435 a girth of at least 15 suces. This implies that the conjecture of Hajós that every graph of chromatic number at least r contains a subdivision of K_r (which is fa ..."
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is false in general) is true for graphs of girth at least 186 (or 15 if r ≥ 436). More generally, we show that for every graph H of maximum degree Δ(H) ≥ 2, every graph G of minimum degree at least max{Δ(H), 3} and girth at least 166 ... contains a subdivision of H
Akademisk avhandling för teknisk doktorsexamen vid
, 1994
"... mcmxciv This thesis deals with combinatorics in connection with Coxeter groups, finitely generated but not necessarily finite. The representation theory of groups as nonsingular matrices over a field is of immense theoretical importance, but also basic for computational group theory, where the group ..."
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the group elements are data structures in a computer. Matrices are unnecessarily large structures, and part of this thesis is concerned with small and efficient representations of a large class of Coxeter groups (including most Coxeter groups that anyone ever payed any attention to.) The main contents
Unavoidable Minors Of Graphs Of Large Type
, 1999
"... . In this paper, we study one measure of complexity of a graph, namely its type. The type of a graph G is defined to be the minimum number n such that there is a sequence of graphs G = G 0 , G1 , : : : , Gn , where G i is obtained by contracting one edge in or deleting one edge from each block of ..."
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of G i\Gamma1 , and where Gn is edgeless. We show that a 3connected graph has large type if and only if it has a minor isomorphic to a large fan. Furthermore, we show that if a graph has large type, then it has a minor isomorphic to a large fan or to a large member of one of two specified families
Graph Minor Theory
 BULLETIN (NEW SERIES) OF THE AMERICAN MATHEMATICAL SOCIETY
, 2005
"... A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory. The motivating problem was Kuratowski’s characterization of planar graphs, and a farreaching ..."
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reaching generalization of this, conjectured by Wagner: If a class of graphs is minorclosed (i.e., it is closed under deleting and contracting edges), then it can be characterized by a finite number of excluded minors. The proof of this conjecture is based on a very general theorem about the structure of large graphs
Results 1  10
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519